2*2 



,1 V.I/ .} ilC GEOMETRY 



[Cm. \i. 



Again, 20 being the angle between the asymptotes, equa- 

 tion [UD] may IK- written 



xy sin 



Fro. 119. 



Now xyzinZO is the area of the parallelogram 

 constructed upon the coordinates of the point /', <>! the 

 hyperbola; and since the coordinates of the vertex A are 



x = y = - , the second member of equation (1) is the 



area of the rhombus ORAS^ constructed upon the coonlinairs 

 of the vertex. Therefore, the area of the parallelogram 

 formed by the asymptotes and lines parallel to them dr 

 from any point of an hyperbola, is constant; it is equal to 

 the rhombus similarly drawn from the vertex of the curve. 

 The equation of the tangent to the hyperbola 



xy = * . . . (2) 



at the point P, is 





The ^-intercept of this tangent is 6>2 7 = 2a; 1 ; hence if OT r 

 be the y-intercept, and M the foot of the ordinate of P v 

 then from the similar triangles MTP l and OTT 1 , 

 TP l : TT' = MTi OT= *! : 2x l = 1 : 2. 



